Eulerian Numbers

I've written a book about some of my favorite topics in enumerative combinatorics. Over the past few decades, Eulerian numbers have arisen in many interesting ways. After introducing a combinatorial definition for Eulerian numbers and deriving some well-known formulas for them, the book makes connections to the study of partially ordered sets, hyperplane arrangements, polytopes, and simplicial complexes. I write about gamma-nonnegativity for Eulerian polynomials, and important partial orderings of the symmetric group: the weak order and the shard intersection order.

The book would be half as long if not for a parallel story of Catalan combinatorics. The Narayana numbers are the central objects here, which possess similar combinatorial and geometric interpretations. Some key objects here include the associahedron and the lattice of non crossing partitions.

The final chapters of the book discuss analogues of the Eulerian and Narayana numbers in the context of Coxeter groups.

My hope is that this book can serve both researchers and graduate students. For example, the first six chapters could work as a text for a one term topics class, while someone interested in the combinatorics of Coxeter groups will find many recent developments collected in the final chapters.